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* The spline calculated during interpolation.
*/
export class CubicInterpolation {
constructor(
private xs: number[],
private ys: number[],
private firstSlope: number,
private lastSlope: number,
private fun: (x: number) => number
) {}
/**
* Returns the value of the function.
*/
public forX(x: number): number {
// queries outside the range of points are matched by linear interpolation
if (x < this.xs[0]) {
return this.ys[0] - this.firstSlope * (this.xs[0] - x);
} else if (x > this.xs[this.xs.length - 1]) {
return (
this.ys[this.ys.length - 1] +
this.lastSlope * (x - this.xs[this.xs.length - 1])
);
} else {
// we use the calculated cubic interpolation
return this.fun(x);
}
}
/**
* Returns the approximate function argument for a given value.
*/
public forY(y: number, precision: number): number {
// queries outside the range of points are matched by linear interpolation
if (y < this.xs[0]) {
return this.xs[0] - (this.ys[0] - y) / this.firstSlope;
} else if (y > this.ys[this.ys.length - 1]) {
return (
this.xs[this.xs.length - 1] +
(y - this.ys[this.ys.length - 1]) / this.lastSlope
);
} else {
// search for the interval y is in, returning the corresponding x if y is one of the original ys
const i = findIntervalIndex(y, this.ys);
if (y === this.ys[i]) {
return this.xs[i];
}
// binary search for X for which Y is within the given precision
let lowX = this.xs[i];
let highX = this.xs[i + 1];
let tries = 50;
while (lowX <= highX && tries > 0) {
const midX = (lowX + highX) / 2;
if (Math.abs(highX - lowX) < precision) {
return midX;
}
const midY = this.forX(midX);
if (Math.abs(midY - y) < precision) {
return midX;
}
if (midY < y) {
lowX = midX;
} else {
highX = midX;
}
tries--;
}
throw new Error(`X was not found for Y = ${y.toString()}`);
}
}
}
/**
* Interpolation of functions based on selected strictly monotonic points.
* Interpolation returns a monotonic spline where each piece is a third-degree polynomial specified in Hermite form.
*
* Method based on https://en.wikipedia.org/wiki/Monotone_cubic_interpolation.
* Method source code inspired by code in https://en.wikipedia.org/wiki/Monotone_cubic_interpolation#Example_implementation.
*/
export const monotoneCubicInterpolation = (
xs: number[],
ys: number[]
): CubicInterpolation => {
const { fun, firstSlope, lastSlope } = createInterpolant(xs, ys);
return new CubicInterpolation(xs, ys, firstSlope, lastSlope, fun);
};
const createInterpolant = (
xs: number[],
ys: number[]
): {
fun: (x: number) => number;
firstSlope: number;
lastSlope: number;
} => {
// checking the initial conditions
if (xs.length != ys.length) {
throw new Error("The number of xs and ys should be equal");
}
if (xs.length === 0) {
throw new Error("Empty array of xs");
}
if (xs.length === 1) {
throw new Error("Interpolation cannot be performed for a single point");
}
// sorting points
const [sortedXs, sortedYs] = sortPoints(xs, ys);
// monotonicity check
if (!isMonotonic(sortedYs)) {
throw new Error("The given points are not monotonic");
}
// consecutive differences and slopes
const dys = [],
dxs = [],
ms = [];
for (let i = 0; i < sortedXs.length - 1; i++) {
const dx = sortedXs[i + 1] - sortedXs[i];
const dy = sortedYs[i + 1] - sortedYs[i];
dxs[i] = dx;
dys[i] = dy;
ms[i] = dy / dx;
}
// degree-1 coefficients
const c1s = [ms[0]];
for (let i = 0; i < dxs.length - 1; i++) {
const m = ms[i],
mNext = ms[i + 1];
Iif (m * mNext <= 0) {
c1s.push(0);
} else {
const dx = dxs[i],
dxNext = dxs[i + 1],
common = dx + dxNext;
c1s.push((3 * common) / ((common + dxNext) / m + (common + dx) / mNext));
}
}
c1s.push(ms[ms.length - 1]);
// degree-2 and degree-3 coefficients
const c2s = [],
c3s = [];
for (let i = 0; i < c1s.length - 1; i++) {
const c1 = c1s[i],
m = ms[i],
invDx = 1 / dxs[i],
common = c1 + c1s[i + 1] - 2 * m;
c2s.push((m - c1 - common) * invDx);
c3s.push(common * invDx * invDx);
}
return {
fun: createInterpolantFunction(sortedXs, sortedYs, c1s, c2s, c3s),
firstSlope: ms[0],
lastSlope: ms[ms.length - 1],
};
};
const sortPoints = (xs: number[], ys: number[]): [number[], number[]] => {
const indexes = [...Array(xs.length).keys()];
indexes.sort((a, b) => {
return xs[a] - xs[b];
});
const sortedXs = [];
const sortedYs = [];
for (let i = 0; i < xs.length; i++) {
sortedXs[i] = xs[indexes[i]];
sortedYs[i] = ys[indexes[i]];
}
return [sortedXs, sortedYs];
};
/**
* Whether the given list of values is monotonic.
*/
const isMonotonic = (ys: number[]) => {
Iif (ys.length < 2) {
return true;
}
// direction of monotonicity
let direction: "increasing" | "decreasing" | "undefined" = "undefined";
let previous = ys[0];
for (let i = 1; i < ys.length; i++) {
const current = ys[i];
// check monotonicity
if (current > previous) {
if (direction === "decreasing") {
return false;
}
direction = "increasing";
} else if (current < previous) {
if (direction === "increasing") {
return false;
}
direction = "decreasing";
}
previous = current;
}
return true;
};
const createInterpolantFunction =
(xs: number[], ys: number[], c1s: number[], c2s: number[], c3s: number[]) =>
(x: number): number => {
// checking whether the argument is from the appropriate range
Iif (x < xs[0] || x > xs[xs.length - 1]) {
throw new Error(
`The function only handles arguments in the range [${xs[0].toString()},${xs[
xs.length - 1
].toString()}]`
);
}
// search for the interval x is in, returning the corresponding y if x is one of the original xs
const i = findIntervalIndex(x, xs);
if (x === xs[i]) {
return ys[i];
}
// interpolate
const diff = x - xs[i];
return ys[i] + diff * (c1s[i] + diff * (c2s[i] + diff * c3s[i]));
};
const findIntervalIndex = (point: number, points: number[]) => {
for (let i = 0; i < points.length - 1; i++) {
if (points[i] <= point && point < points[i + 1]) {
return i;
}
}
if (point === points[points.length - 1]) {
return points.length - 1;
}
throw new Error(
`The point (${point}) is outside the point list [${points.join(",")}]`
);
};
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